Section 5.1 – Integers: Addition and Subtraction
How may we compare the values of the following two accounts?
Sam had two accounts, a savings account and a loan. She had a certificate of deposit worth $4,360 and a car loan of
$8,290. Which account has the greatest value?
The answer to this question depends on how we think about the situation. If we are asking which situation involves more
dollars, then $8,290 involves a greater dollar amount than $4,360. However, if we are asking which value is worth more to Sam,
obviously the certificate of deposit, which is money she “owns” is worth more than the loan, which is money she “owes”. In
mathematical terms, Sam’s CD has a positive value and the loan has a negative value. In mathematics, when we look at just the
number of dollars and not whether the value is positive or negative we are talking about the absolute values.
Definition. The set of integers is the set {…, –3, –2, –1, 0, 1, 2, 3, …}.
Notice that this is the set of whole numbers {0, 1, 2, 3, 4, . . . } together with all of the opposites. (The opposite of zero is
zero.)
We consider two different models for the set of integers, {…, –3, –2, –1, 0, 1, 2, 3, …}.
1.
The chip model uses colored beans to represent negative integers and positive integers.
A green bean or shaded circle,
, is used to represent 1.
A red bean or open circle,
, is used to represent –1.
Examples:
2.
(a)
(b)
(c)
(d)
A deposit of $1 may be represented by
.
A withdrawl of $1 may be represented by
.
Receiving 3 candy bars may be represented as
Giving away 3 candy bars may be represented as
.
.
The number-line model (or measurement model) uses vectors along a number line to represent negative integers and
positive integers.
A vector pointing to the right will be used to represent 1.
1
–5 –4 –3
–2
–1
0
1
2
3
4
5
A vector pointing to the left will be used to represent –1.
–1
–5 –4 –3
Examples:
(a)
(b)
(c)
(d)
–2
–1
0
1
2
3
4
5
One step forward, +1.
One step backware, –1.
A temperature of one degree above zero, +1.
A temperature of one degree below zero, –1.
We could write the value positive 1 as +1, but we normally leave out the positive sign. A value with no sign is assumed to be
a positive value.
Comparing the Two Models
The number line is a good model for the expressing positions relative to some specific value, such as graphing on coordinate
planes (relative to the origin), temperatures on thermometers (relative to zero degrees or to freezing), and elevation (relative to sea
level). It is also good when we need to express fractional parts of the whole, since we can find these on the number line, but it
would be difficult to do with beans or chips.
The chip model is a good model for small amounts (with no fractional parts) and for items that have atomic charges such as
protons, electrons, and neutrons. It is also a helpful model for understanding how integer arithmetic works.
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Representations of Zero in the Chip Model
Another important concept with the chip model is that there are many different ways to represent an amount of zero. For
example, a deposit $1, represented by
, into a new account followed by a withdrawal of $1, represented by
, from the
account would leave an account balance of $0. Thus zero would be represented by
, though it would be better to remove
both beans. In other words, beans of opposite color cancel each other out, so both beans could be removed. This example
illustrates the property that 1 + (–1) = –1 + 1 = 0.
More Examples:
(a) No chips at all would represent an amount of zero
(b)
would represent an amount of zero, e.g., a deposit of $1 and a withdrawal of $1 leaves no change in the
balance.
(c)
would represent an amount of zero, e.g., receive 3 candy bars and give away 3 candy bars
leaves no change in the amount of candy bars.
In other words, two chips of opposite color cancel each other out, so both chips can be removed. This example illustrates the
property that 1 + (–1) = –1 + 1 = 0.
We generalize this relationship with examples illustrated with both models.
3 + (–3) = 0
–5 –4 –3
–2
–1
0
–5 + 5 = 0
–3
5
3
–5
1
2
3
4
5
–5 –4 –3
Beginning at zero, in the measurement model, a move
of three units to the right followed by three units to the
left brings us back to zero.
–2
–1
0
1
2
3
4
5
Beginning at zero, in the measurement model, a move
of five units to the left followed by five units to the
right brings us back to zero.
We notice that the result of combining a number and its opposite is zero. This motivates the following formal definition.
Definition. For every integer n, there is a unique integer m such that n + m = m + n = 0. The integer m is called the additive inverse
of n. This property of integers is called the inverse property for integer addition.
Examples:
Find the additive inverse for each of the following integers.
Integer
7
–5
–13
29
0
Additive Inverse
–7
5
13
–29
0
Illustrate the additive inverse for each of the integers above with the measurement model.
If we consider the negative sign as representing the opposite, the above examples illustrate that
–(–5) = 5
–(+7) = –7
–(–13) = 13
–(+29) = –29
–(0) = 0
that is, the opposite of an integer is equivalent to the additive inverse of the integer.
Explain how the chip model may be used to illustrate the previous examples, i.e., show the additive inverse of –4 is 4.
That is show –(–4) = 4.
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Absolute Value
Suppose Pat walked 5 steps forward and Kim walked 6 steps backward. Who took more steps?
Even though we would represent Pat’s steps as 5 and Kim’s steps as –6, we would say that Kim took more steps. This
problem and the opening problem for this information sheet both motivate the concept of absolute value.
Absolute value refers to the distance from zero without reference to direction. Absolute value, since it is a non-directed
distance, is always zero or positive.
For the above example: |5| read as the “absolute value of 5” is 5 since it is five units from 0 (the origin).
5
–5 –4 –3
–2
–1
0
1
2
4
3
5
|–6| read as the “absolute value of –6” is 6 since it is six units from 0 (the origin).
–6
–6 –5 –4
–3
–2
–1
0
1
3
2
4
 a if a  0
.
 a if a  0
Formal Definition of Absolute Value: The absolute value of a is defined by a  
Note the way the absolute value of a negative integer is evaluated by use of the definition, e.g., |–7| = –(–7) = 7.
Examples:
Find the absolute value for each of the following integers.
Integer
7
–5
–13
29
0
Absolute Value
7
5
13
29
0
Integer Addition
Motivation Problem
Sam has two accounts a savings account and a loan. The savings account is a certificate of deposit worth $4,360 and the
loan is a car loan of $8,290. What is the total value of these two accounts?
The question is asking us to sum the two values where one value is positive and the other value is negative, specifically to
find 4,360 + (–8,290). If we think of the problem as using the money in the CD to pay down the car loan, we can consider this as a
subtraction problem where the loan is reduced by $4,360 so that the new loan value would be $3,930. This implies that 4,360 + (–
8,290) = –3,930.
Note that the same symbol “+“ is used for addition and positive. When used for addition, the symbol “+“ is called a plus
sign; whereas, when used to represent a certain direction , the symbol “+“ is called a positive sign. In this course, we distinguish
between plus (to add) and positive (direction).
Addition of Integers – When Both Values are Positive
Use both the chip model and the number-line model to illustrate and solve each problem.
(Make sure you do each problem physically (chip model) by using the green and red colored beans.)
2+3=
1+3=
Make up at least two more problems for the addition of two integers that have the same sign.
Write a rule, which describe your observations, for adding two positive integers.
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Here are two illustrations.
2+3=5
1+3=4
Chip Model:
Chip Model:
Number Line Model:
Number Line Model:
3
2
–5 –4 –3
–2
–1
0
1
2
3
1
4
–5 –4 –3
5
–2
–1
0
3
1
2
3
4
5
2
3
4
5
Summary: The sum of the two positive integers is the same as adding two whole numbers.
Addition of Integers – When Both Values are Negative
Use both the chip model and the number-line model to illustrate and solve each problem.
(Make sure you do each problem physically (chip model) by using the green and red colored beans.)
–2 + (–3) =
–1 + (–3) =
Make up at least two more problems for the addition of two integers that have the same sign.
Write a rule, which describe your observations, for adding two positive integers.
Here are two illustrations.
–3 + (–2) = –5
–2
–2 + (–1) = –3
–3
–5 –4 –3
–2
–1
–1
0
1
2
3
4
–5 –4 –3
5
–2
–2
–1
0
1
Summary: The sum of two negative integers is the sum of the absolute values and keeping the negative sign.
More Examples:
(a)
14 + 28 = 42
(b)
–124 + (–79) = –203
(c)
–57 + (–36) = –93
(d)
58 + 32 = 90
Addition of Integers – When the Signs are Different
Use both the chip model and the number-line model to illustrate and solve each problem.
(Make sure you do each problem physically (set model) by using the green and red colored beans.)
2 + (–3) =
–1 + 3 =
4 + (–1) =
–5 + 3 =
Make up at least two more problems for the addition of two integers that have different signs.
Write one or two rules, which describe your observations, for adding two integers with opposite signs.
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is equivalent to zero, i.e., –1 + 1 = 0.
A reminder from the previous section, when using the set model
2 + (–3) = –1
–1 + 3 = 2
–3
3
–1
2
–5 –4 –3
–2
–1
0
1
2
3
4
–5 –4 –3
5
–2
–1
–1
0
1
4
3
5
2
3
–5
–1
4
–2
2
–5 + 3 = –2
4 + (–1) = 3
–5 –4 –3
1
0
4
3
5
–5 –4 –3
–2
–1
0
1
2
3
4
5
Summary: The sum of two integers with different signs is the difference between the absolute values (the greater absolute value
minus the lesser or same absolute value) with the sign of the integer that has the greatest absolute value.
More Examples:
(a)
14 + (–27) = –13
(b)
–124 + 79 = –45
(c)
57 + (–36) = 21
(d)
–32 + 58 = 26
The above examples and illustrations motivate the following rules for determining the sum of any two integers.
Sign Rules for Adding Two Integers
1.
If the two integers have the same sign, find the sum of the absolute values of the two integers and use the same sign as
the two integers.
a + b = sign(a) (|a| + |b|)
2.
If the two integers have different signs, find the difference between the absolute values of the two integers and use the
sign of the integer with the greatest absolute value.
Assume |a| > |b|, then a + b = sign(a) (|a| – |b|).
Properties of Integers
State the property of integer addition.
(a) 3 + (–5) is an integer
(d) –8 + (–5) = –5 + (–8)
(b) [2 + (–1)] + (–4) = 2 + [(–1) + (–4)]
(e) 0 is the unique integer such that
–3 + 0 = 0 + (–3) = –3.
(c) –7 is the unique integer such that 7 + (–7) = 0.
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Properties of Integer Addition
1.
Closure Property for Integer Addition.
Let a and b be any integers. Then a + b is a unique integer.
2.
Commutative Property of Integer Addition.
Let a and b be any integers. Then a + b = b + a.
3.
Associative Property for Integer Addition.
Let a, b, and c be any integers. Then a + (b + c) = (a + b) + c.
4.
Identity Property for Integer Addition.
Let a be any integer. Then a + 0 = 0 + a = a. The integer 0 is called the additive identity.
5.
Inverse Property for Integer Addition.
For every integer a, there is a unique integer b such that a + b = b + a = 0. The integer b is called the additive inverse of
a.
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Integer Subtraction
Motivation Problem
Sam had two accounts, a savings account and a loan. She had a certificate of deposit worth $4,360 and a car loan of
$8,290. What is the difference between the values of these two accounts?
The question is asking us to find the difference in the values where one value is positive and the other value is negative,
4,360 – (–8,290). If we think of the problem as how much money would Sam need to have to pay off the loan and keep the CD, we
may consider it as an addition problem where the 8,290 is added to the 4,360 so that we would need $12,650 to pay off the loan
and still have the CD. This implies that 4,360 – (–8,290) = 4,360 + 8,290 = 12,650.
Note that the same symbol “–“ is used for subtraction and opposite. When used for subtraction, the symbol “–“ is called a
minus sign; whereas, when used for the opposite, the symbol “–“ is called a negative sign. In this course, we distinguish between
minus (to remove) and negative (opposite).
Review. There are two methods that were used for the subtraction of whole numbers: the take-away approach and the missing
addend approach. We use these two approaches as we explore several cases of subtraction of two integers with the chip model (set
model) and the number line model.
Missing Addend Approach. When we studied whole number subtraction, we learned that for whole numbers a and b, a – b = c if
and only if there is a whole number c such that b + c = a. The a is called the minuend, the b is called the subtrahend and the c is
called the difference.
Subtraction of Integers — When the Signs are the Same and the Absolute Value of the Minuend is Greater Than the Absolute
Value of the Subtrahend.
Notice that when the two integers are both positive and the subtrahend is less than the minuend, the problem is equivalent to
subtracting two whole numbers, e.g., 8 – 5 = 3. Also, the result is equivalent to adding the opposite of the subtrahend, e.g., 8 – 5 =
8 + (–5) = 3.
Further, the same holds for when both integers are negative and the subtrahend is greater than the minuend as seen with the
following example using the take-away chip model with the problem –5 – (–3).
–5 – (–3) = –2
Also, note that the result is equivalent to adding the opposite of the subtrahend, e.g.,
–5 – (–3) = –5 + 3 = –2.
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Subtraction of Integers — When the Signs are the Different.
We use three methods/models for subtracting two integers (Pattern, Take-Away, and Missing Addend).
Next, we subtract a negative integer from a positive integer. Consider the problem
3 – (–2).
Pattern
Take-Away Approach
Missing Addend Approach
3 – (–2) = N iff –2 + N = 3
3 – 3 =0
3 – 2 =1
3 – 1 =2
3 – 0 =3
3 – (–1) = 4
3 – (–2) = 5
–2
N=5
–2
3
0
Since we are familiar with whole
number subtraction, we begin with
whole number problems (positive
integers) for which we know the
solutions and then establish a pattern
into the negative integers.
1
2
3
4
5
Since we do not have –2 units that can
be taken from 3, we add 0 where 0 = 2
+ (–2). Now we have –2 units that can
be taken away. Using the properties
for integer addition, the procedure can
be written as
3 – (–2)
= [3 + 0] – (–2)
= [3 + 2 + (–2)] – (–2)
= [3 + 2] + [(–2) – (–2)]
= [3 + 2] + 0
= 3 + 2 = 5.
–2
–1
0
1
2
3
We ask ourselves the question: What
do we need to add to –2 units to
obtain a result of 3 units? We need to
add 5 units to –2 units to obtain 3
units.
For the above case, it appears that we can subtract two integers by adding the opposite of the subtrahend to the minuend,
i.e., 3 – (–2) = 3 + 2 = 5.
State the property of integer addition which justifies each step in the take-away approach.
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Next, we subtract a positive integer from a negative integer. Consider the problem
–3 – 2.
Pattern
Take-Away Approach
Missing Addend Approach
5–2 = 3
4–2 = 2
3–2 = 1
2–2 = 0
1 – 2 = –1
0 – 2 = –2
–1 – 2 = –3
–2 – 2 = –4
–3 – 2 = –5
–3 – 2 = N iff 2 + N = –3
N = –5
2
–5
Since we are familiar with whole
number subtraction, we begin with
whole number problems for which we
know the solutions and then establish
a pattern into the negative integers.
–3
–4 –3 –2
–1
2
0
Since we do not have 2 units that
can be taken from –3 units, we add 0
where 0 = –2 + 2. Now we have 2
units that can be taken away. Using
the properties for integer addition, the
procedure can be written as
–3 – 2
= [–3 + 0] – 2
= [–3 + (–2) + 2] – 2
= [–3 + (–2)] + [2 – 2]
= [–3 + (–2)] + 0
= –3 + (–2) = –5.
–3 –2
–1
0
1
2
We ask ourselves the question:
What do we need to add to 2 units to
obtain a result of –3 units? We need to
add –5 units to 2 units to obtain –3
units.
For the above case, it appears that we can subtract two integers by adding the opposite of the subtrahend to the minuend,
i.e., –3 – 2 = –3 + (–2) = –5.
State the property of integer addition which justifies each step in the take-away approach.
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Subtraction of Integers — When the Signs are the Same.
Next, we subtract two negative integers where the subtrahend is less than the minuend (or the absolute value of the subtrahend
is greater than the absolute value of the minuend). Consider the problem –3 – (–5).
–3 – (–5)
Pattern
Take-Away Approach
Missing Addend Approach
–3 – (–5) = N iff –5 + N = –3
–3 – 0 = –3
–3 – (–1) = –2
–3 – (–2) = –1
–3 – (–3) = 0
–3 – (–4) = 1
–3 – (–5) = 2
2
–2
–3
–3 –2
Two key points in the pattern are –
3 – 0 = –3 and
–3 – (–3) = 0. Further, we use the
earlier results for subtracting two
integers where the absolute value of
the subtrahend is less than the
minuend.
–1
0
1
N=2
–5
2
Since we do not have –5 units that
can be taken from –3 units, we add 0
where 0 = –2 + 2. Now we have –5
units that can be taken away. Using
the properties for integer addition, the
procedure can be written as
–3 – (–5)
= [–3 + 0] – (–5)
= [–3 + (–2) + 2] – (–5)
= [–3 + (–2)] + 2 – (–5)
= [–5 – (–5)] + 2
= 0 + 2 = 2.
–5
–4 –3 –2
–1
0
We ask ourselves the question:
What do we need to add to –5 units to
obtain a result of –3 units? We need to
add 2 units to –5 units to obtain –3
units.
Again it appears that we can subtract two integers by adding the opposite of the subtrahend to the minuend, i.e., –3 – (–5)
= –3 + 5 = 2.
State the property of integer addition which justifies each step in the take-away approach.
More Examples. Directions. Use the three methods (pattern, take-away, and missing addend) to solve each problem. For the takeaway and missing addend approaches, use both the set model and the measurement model to illustrate and solve each problem.
(Make sure you do each problem physically (set model) by using the green and red
colored chips.)
–3 – (–2)
–3 – 2
3–2
Explain how the take-away approach shows that subtraction of integers may be computed by the method of Adding the Opposite.
That is 3 – (–2) = 3 + 2 = 5. Use the set model and number line model to illustrate this method for each of the previous four
problems.
Which method do you find the easiest to compute with? (Pattern, Take-Away, Missing Addend, or Adding the
Opposite) Explain.
Rule for Subtracting Two Integers. The difference between two integers may be found by adding the opposite of the subtrahend
to the minuend, i.e., a – b = a + (–b).
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Examples: Write each problem as a mathematical expression and compute the solution. Always finish application problems by
writing the solution in a complete sentence as demonstrated in these examples.
1.
Kim had a score of 6 points in a game of cards, but 2 points had been deducted that should not have been deducted.
What should Kim’s score be?
6 – (–2) = 6 + 2 = 8
Kim’s score should be 8 points.
2.
The temperature yesterday was 14° below zero and the temperature today is 6° below zero. What is the change in
temperature from yesterday to today?
–6 – (–14) = –6 + 14 = 8
The temperature increased by 8° from yesterday to today.
3.
Acme, Inc. earned $5 million and paid out $6 million. What is Acme, Inc.’s net profit?
5 – 6 = 5 + (–6) = –1
Acme, Inc. had a net loss of $1 million.
4.
In a game of Captain May I, Pat had walked 4 steps backward followed by 3 steps backward. What as the net change
in Pat’s position?
–4 + (–3) = –7
Pat moved a total of seven steps backwards.
Problems and Exercises.
1.
Symbolize each statement with the appropriate positive or negative rational number.
(a)
An excess of 2.4 inches of rainfall
(b)
The temperature is 6° below normal.
(c)
A downward force of 84 pounds
(d)
A latitude of 75° north
(e)
An inventory shortage of 45 items
(f)
A 2-point increase in stock value
(g)
A liability of $2,478.39
(h)
A charge of 6 amperes of electricity
(i)
A debt of $4,580
(j)
A temperature of 5° below zero
(k)
Walked 8½ feet forwards
(l)
The lowest point in the United States 282 feet
below sea level is in Death Valley.
(m) A checking account balance of $578.36
(n)
Pat’s credit card balance is $794.21.
(o)
Eighteen days ago
(p)
Fall of Rome in 476 A.D.
(q)
A loss of $60
(r)
The cost of living increased by 3%
2.
Write an expression or statement that is the opposite of each part of Exercise 1 and symbolize with the appropriate positive or
negative rational number.
3.
Simplify the following to a single integer value.
(a)
4.
– (–12)
(b) –|– 9|
(c) – (– (– (– (2))))
(d) – |– (–4)|
Let N, W, and I represent the natural numbers, whole numbers, and integers, respectively. Identify whether each statement is
true or false. Further, if the statement is false, give a counterexample, i.e., an example of when the statement is not true.
(a)
W is a subset of N.
(b)
I is a subset of N.
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(c) N is a subset of W.
For each of the following problems, write as a mathematical expression, compute the answer, and write the solution in a
complete sentence.
5.
The temperature was 4 below zero at 11:00 and rose 5 the next hour. What was the temperature at 12:00?
6.
The temperature was 9 below zero at 9:00 and dropped 7 the next hour. What was the temperature at 10:00?
7.
The temperature was 14 below zero yesterday and the temperature is 8 below zero today. What is the change in temperature
from yesterday to today?
8.
The temperature was 7 below zero yesterday and the temperature is 15 below zero today. What is the change in temperature
from yesterday to today?
9.
Lynn has two bank accounts with balances of $1100 and $843. Lynn also has three credit cards with balances of $250, $782,
and $1360. What is the total value of the five accounts?
10. Pat had a credit card balance of $492. If Pat paid $275, what would be the new credit card balance?
11. An error was made by the credit card company on Kim's account. The credit card statement showed that Kim owed a total of
$353, but a charge of $45 was incorrectly put on the statement. What is Kim's correct balance?
12. While playing cards, Sam was 14 points in the hole. If Sam lost eight points on the next hand, what would be Sam's new
score?
13. In a game of Captain-May-I, you had taken 7 steps forward, 9 steps backward, 4 steps backward, 2 steps forward, and 3 steps
backward. Where are you now?
14. In a series of downs, a football team gained 7 yards, lost 4 yards, and lost 2 yards. The team was then penalized 10 yards and
on the following play the opposing team was penalized 5 yards. The team then gained 4 yards. What was the total gain or
loss?
15. In a week, a given stock gained 5 points, dropped 12 points, dropped 3 points, gained 18 points, and dropped 10 points. What
was the net change in the stock's worth?
16. Fred paid $38 for an item and Sally paid $31 for a similar item. How much more or less did Fred pay than Sally paid?
17. Maala made several deposits and withdrawals from a checking account. Maala deposited $50, wrote a check for $83,
deposited $45, and wrote a check for $31. What is the total change in her checking account?
18. Joe thought he owed Mandy $27, but actually owed Mandy $39. How much more or less did Joe actually owe Mandy?
19. What day is the day before two days after yesterday?
20. The account balances for Lynn’s seven accounts are given in the table below. Find the total value of Lynn’s seven accounts.
Accounts
Checking
C.D
Mortgage
Credit Card
401 K
Roth IRA
Car Loan
Value
$1,429
$12,200
$125,005
$3,070
$97,592
$17,180
$9,047
21. Make up at least four integer addition word problems, one for each of the four types of problems.
22. Make up at least four different types of integer subtraction word problems.
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